Robotics Research Technical Report Characterization of Signals From Multiscale Edges (Classic Reprint)
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Opis: Robotics Research Technical Report - Mallat Stephane

Excerpt from Robotics Research Technical Report: Characterization of Signals From Multiscale Edges
Points of sharp variations are often among the most important features for analyzing the properties of transient signals or images. In images, they are generally located at the boundaries of important image structures. In order to detect the contours of small structures as well as the boundaries of larger objects, several researchers in computer vision have introduced the concept of multiscale edge detection [24, 29, 31]. The scale defines the size of the neighborhood where the signal changes are computed. The wavelet transform is closely related to multiscale edge detection and can provide a deeper understanding of these algorithms. We concentrate on the Canny edge detector [3[, which is equivalent to finding the local maxima of a wavelet transform modulus.
There are many different types of sharp variation points in images. Edges created by occlusions, shadows, highlights, roofs, textures. have very different local intensity profiles. To label more precisely an edge that has been detected, it is necessary to analyze its local properties. In mathematics, singularities are generally characterized by their Lipschitz exponents. The wavelet theory proves that these Lipschitz exponents can be computed from the evolution across scales of the wavelet transform modulus maxima. We derive a numerical procedure to measure these exponents. If an edge is smooth, we can also estimate how smooth it is from the decay of the wavelet transform maxima across scales. Lipschitz exponents and smoothing factors arc numerical descriptors that allow us to discriminate the intensity profiles of different types of edges.
An important open problem in computer vision is to understand how much information is carried by multiscale edges and how stable is a multiscale edge representation. This issue is important in pattern recognition where one needs to know whether some interesting information is lost when representing a pattern with edges. We study the reconstruction of one and two-dimensional signals from multiscale edges detected by the wavelet transform modulus maxima. It has been conjectured [22, 24] that multiscale edges characterize uniquely one and two-dimensional signals, but recently Meyer [27] has found counter-examples to these conjectures. In spite of these counter-examples, we show that one can reconstruct a close approximation of the original signal from multiscale edges. The reconstruction algorithm is based on alternate projections. We prove its convergence and derive a lower bound for the convergence rate. Numerical results are given both for one and two-dimensional signals. The differences between the original and reconstructed images are not visible on a high quality video monitor.
The ability to reconstruct images from multiscale edges has many applications in signal processing. It allows us to process the image information with edge based algorithms. We describe a compact image coding algorithm that keeps only the "important" edges. The image that is recovered from these main features has lost some small details but is visually of good quality. Examples with compression ratio over 30 are shown.
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Szczegóły: Robotics Research Technical Report - Mallat Stephane

Tytuł: Robotics Research Technical Report
Podtytuł: Characterization of Signals From Multiscale Edges (Classic Reprint)
Autor: Mallat Stephane
Wydawnictwo: FB &c Ltd
ISBN: 9781332111657
Języki: angielski
Rok wydania: 2015
Ilość stron: 80
Format: 15.2x22.9cm
Oprawa: Miękka


Recenzje: Robotics Research Technical Report - Mallat Stephane
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Robotics Research Technical Report
Characterization of Signals From Multiscale Edges (Classic Reprint)

Excerpt from Robotics Research Technical Report: Characterization of Signals From Multiscale Edges
Points of sharp variations are often among the most important features for analyzing the properties of transient signals or images. In images, they are generally located at the boundaries of important image structures. In order to detect the contours of small structures as well as the boundaries of larger objects, several researchers in computer vision have introduced the concept of multiscale edge detection [24, 29, 31]. The scale defines the size of the neighborhood where the signal changes are computed. The wavelet transform is closely related to multiscale edge detection and can provide a deeper understanding of these algorithms. We concentrate on the Canny edge detector [3[, which is equivalent to finding the local maxima of a wavelet transform modulus.
There are many different types of sharp variation points in images. Edges created by occlusions, shadows, highlights, roofs, textures. have very different local intensity profiles. To label more precisely an edge that has been detected, it is necessary to analyze its local properties. In mathematics, singularities are generally characterized by their Lipschitz exponents. The wavelet theory proves that these Lipschitz exponents can be computed from the evolution across scales of the wavelet transform modulus maxima. We derive a numerical procedure to measure these exponents. If an edge is smooth, we can also estimate how smooth it is from the decay of the wavelet transform maxima across scales. Lipschitz exponents and smoothing factors arc numerical descriptors that allow us to discriminate the intensity profiles of different types of edges.
An important open problem in computer vision is to understand how much information is carried by multiscale edges and how stable is a multiscale edge representation. This issue is important in pattern recognition where one needs to know whether some interesting information is lost when representing a pattern with edges. We study the reconstruction of one and two-dimensional signals from multiscale edges detected by the wavelet transform modulus maxima. It has been conjectured [22, 24] that multiscale edges characterize uniquely one and two-dimensional signals, but recently Meyer [27] has found counter-examples to these conjectures. In spite of these counter-examples, we show that one can reconstruct a close approximation of the original signal from multiscale edges. The reconstruction algorithm is based on alternate projections. We prove its convergence and derive a lower bound for the convergence rate. Numerical results are given both for one and two-dimensional signals. The differences between the original and reconstructed images are not visible on a high quality video monitor.
The ability to reconstruct images from multiscale edges has many applications in signal processing. It allows us to process the image information with edge based algorithms. We describe a compact image coding algorithm that keeps only the "important" edges. The image that is recovered from these main features has lost some small details but is visually of good quality. Examples with compression ratio over 30 are shown.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

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Szczegóły: Robotics Research Technical Report - Mallat Stephane

Tytuł: Robotics Research Technical Report
Podtytuł: Characterization of Signals From Multiscale Edges (Classic Reprint)
Autor: Mallat Stephane
Wydawnictwo: FB &c Ltd
ISBN: 9781332111657
Języki: angielski
Rok wydania: 2015
Ilość stron: 80
Format: 15.2x22.9cm
Oprawa: Miękka


Recenzje: Robotics Research Technical Report - Mallat Stephane

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